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Algebraicity and implicit definability in set theory, CUNY, May 2013

Abstract. An element a is definable in a model M if it is the unique object in M satisfying some first-order property. It is algebraic, in contrast, if it is amongst at most finitely many objects satisfying some first-order property φ, that is, if { b | M satisfies φ[b] } is a finite set containing a. In this talk, I aim to consider the situation that arises when one replaces the use of definability in several parts of set theory with the weaker concept of algebraicity. For example, in place of the class HOD of all hereditarily ordinal-definable sets, I should like to consider the class HOA of all hereditarily ordinal algebraic sets. How do these two classes relate? In place of the study of pointwise definable models of set theory, I should like to consider the pointwise algebraic models of set theory. Are these the same? In place of the constructible universe L, I should like to consider the inner model arising from iterating the algebraic (or implicit) power set operation rather than the definable power set operation. The result is a highly interesting new inner model of ZFC, denoted Imp, whose properties are only now coming to light. Is Imp the same as L? Is it absolute? I shall answer all these questions at the talk, but many others remain open.

3 thoughts on “Algebraicity and implicit definability in set theory, CUNY, May 2013”

Oh, the talk was a lot of fun, and everything went fine, except for one unfortunate and embarrassing thing: while discussing our claim that the Imp of Imp is Imp, which was used to get AC holding in Imp, I realized in the middle of presenting the proof that the argument we had used seems not actually to work fully; it has a subtle hole. I’m not yet sure how to fix this issue. The rest of the results were fine, but this one part exploded mid-talk!

It is interesting that Imp can think it is Hod, but can’t actually be Hod since Hod cannot think, I think this is how it goes, that the universe is Hod. Though, you showed that Imp more like L in this way since it thinks it is the universe. How nice, this Imp, somewhere between Hod and L. Thank you for your talk, good luck on the “explosion”!